Summary: Multiscale morphology is useful for many image analysis tasks. A compact representation for multiscale dilations and erosions of binary images is the distance transform. It can be implemented either via using a purely discrete approach of discrete modeling and discrete processing or via a continuous modeling of the problem and then discrete processing. The discrete approach uses chamfer metrics which yield multiscale dilations/erosions of the image by polygonal structuring elements. The chamfer distance transform is an approximation of the Euclidean distance transform, and the approximation error increases with the scale. To achieve a better approximation of the true Euclidean geometry at large scales, a continuous approach can be used where multiscale dilations/erosions by disks are modeled as solutions of PDEs running for a time period equal to the scale. In this paper, we compare the discrete approach of the chamfer distance transform with the continuous approach of the morphological PDEs implemented via numerical algorithms of curve evolution. We find that, for binary images, the chamfer distance transform is easier to implement and should be used for small scale dilations/erosions. Implementing the distance transform via curve evolution is more complex, but at medium and large scales gives a better and very close approximation to the true Euclidean geometry. For gray-level images, curve evolution achieves better accuracy than gray-weighted chamfer distances in implementing the gray-weighted distance transform which is an approximate solution of the eikonal PDE encountered in vision applications.
For the entire collection see [Zbl 0885.00059].